AP® Precalculus · Unit 3 · Topic 3.3
Exact Values from Special Triangles
Special right triangles let you compute exact sine and cosine values at 30, 45, and 60 degrees instead of decimal approximations, because those angles come from cutting an equilateral or square shape in half.
Most angles give sine and cosine values that are messy infinite decimals. The angles 30, 45, and 60 degrees are different because you can build them from shapes with known side ratios. Cut an equilateral triangle with side length 2 straight down the middle and you get two 30-60-90 triangles: the base is 1, the height is √3, and the hypotenuse is 2. Read the ratios off those sides and you get sin(30 degrees) = 1/2 and cos(30 degrees) = √3/2, exactly, no rounding. Do the same trick with a square of side 1, sliced along its diagonal, and you get a 45-45-90 triangle with legs 1 and hypotenuse √2, which hands you cos(π/4) = sin(π/4) = √2/2.
Once you have those ratios, the unit circle turns them into coordinates. The point at π/6 is (√3/2, 1/2), at π/4 it's (√2/2, √2/2), and at π/3 it's (1/2, √3/2) — cosine and sine just swap roles as the angle steepens. These three triangles are the only ones that hand you exact values this cleanly, which is why they anchor every other angle on the circle.
Angles outside the first quadrant borrow these same numbers through their reference angle, the acute angle to the nearest x-axis. An angle in the third quadrant, say 7pi/6, has reference angle π/6, so its sine and cosine have the same magnitudes as π/6's — 1/2 and √3/2 — but both flip negative, since every point in the third quadrant sits in the region where x and y are both negative. This is the bridge into reading sine and cosine as coordinate functions all the way around the circle, not just facts about triangles.
Key terms
- Reference angle
- The acute angle formed between the terminal side of an angle and the x-axis.
- 30-60-90 triangle
- A right triangle from half an equilateral triangle, with side ratios 1, √3, 2.
- 45-45-90 triangle
- A right isosceles triangle from half a square, with side ratios 1, 1, √2.
- Unit circle
- A circle of radius 1 centered at the origin, where each point's coordinates are (cos(theta), sin(theta)).
Watch it explained
- How it worksHow does slicing an equilateral triangle in half produce the exact value of sin(30 degrees)?
- How it worksWalk me through the isosceles right triangle that gives cos(π/4) its exact value.
- Worked exampleFind the exact coordinates of the unit circle points at π/6, π/4, and π/3.
- Going furtherWhat happens to those exact values once the angle lands in the third quadrant?
Check yourself
End of topic.